Fox, Ronald Forrest, and George E. Uhlenbeck. “Contributions to non‐equilibrium thermodynamics. I. Theory of hydrodynamical fluctuations.” The Physics of Fluids 13, no. 8 (1970): 1893-1902.
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The velocity of a particle in Brownian motion as described by the Langevin equation is a stationary Gaussian–Markov process. Similarly, in the formulation of the laws of non‐equilibrium thermodynamics by Onsager and Machlup, the macroscopic variables describing the state of a system lead to an 𝑛n‐component stationary Gaussian–Markov process, which, in addition, these authors assumed to be even in time. By dropping this assumption, the most general stationary Gaussian–Markov process is discussed. As a consequence, the theory becomes applicable to the linearized hydrodynamical equations and suggests that the Navier–Stokes equations require additional fluctuation terms which were first proposed by Landau and Lifshitz. Such terms and their correlation properties are presented, and these equations are then used to derive the Langevin equation for the Brownian motion of a particle of arbitrary shape.